Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 16, 2012; final manuscript received September 22, 2013; published online November 7, 2013. Assoc. Editor: Luis Alvarez.

Pendulum models have been studied as benchmark problems for development of nonlinear control schemes, as well as reduced-order models for the dynamics analysis of gait, balance and fall for humanoid robots. We have earlier introduced the reaction mass pendulum (RMP), an extension of the traditional inverted pendulum models, which explicitly captures the variable rotational inertia and angular momentum of a human or humanoid. The RMP consists of an extensible “leg”, and a “body” with moving proof masses that gives rise to the variable rotational inertia. In this paper, we present a thorough analysis of the RMP, which is treated as a three-dimensional (3D) multibody system in its own right. We derive the complete kinematics and dynamics equations of the RMP system and obtain its equilibrium conditions. We show that the equilibria of this system consist of an unstable equilibrium manifold and a stable equilibrium manifold. Next, we present a nonlinear control scheme for the RMP, which is an underactuated system with three unactuated degrees of freedom (DOFs). This scheme asymptotically stabilizes this underactuated system at its unstable equilibrium manifold, with a vertically upright configuration for the “leg” of the RMP. The domain of convergence of this stabilization scheme is shown to be almost global in the state space of the RMP. Numerical simulation results verify this stability property of the control scheme and demonstrate its effectiveness in stabilizing the unstable equilibrium manifold.

References

Figures

This figure illustrates the main difference between the traditional point-mass inverted pendulum model with zero rotational inertia (left) and models containing nonzero rotational inertia (right). The point mass in the traditional pendulum model forces the ground reaction force, f, to pass through the center of mass. A reaction mass type pendulum, by virtue of its nonzero rotational inertia, allows the ground reaction force to deviate from the lean line.

Conceptual realization of the 3D RMP. The RMP consists of a telescopic leg connecting the CoP and CoM, and a rotating body. In 3D, the rotating ellipsoidal body is mechanically equivalent to six equal masses on three mutually perpendicular rotating tracks. The CoM of the RMP is fixed at the common midpoint of the tracks. This is the point about which the tracks themselves can rotate.

Three coordinate frames are used to describe the RMP kinematics: the CoP-fixed inertial frame {I}, the leg-fixed frame {L} located at the CoM and aligned with the leg axis, and the PMA assembly-fixed frame {P} located at the CoM. ρ is the length of the RMP leg, RL is the rotation matrix from {L} to {I}, RPL is the rotation matrix from {P} to {L}, and s=[s1 s2 s3]T is the vector of locations of the three pairs of proof mass actuators in {P}.

Time plot of the angle Φ between the reduced attitude ΓL=RLTe3 of the RMP leg and the opposite direction to uniform gravity −e3, for the RMP system with the feedback control scheme of Theorem 1. Here, Φ = 48.6 deg at the start of the simulation and the final desired value of Φ is 0 deg.

Time plots of the components of the angular velocity ΩL of the RMP leg, for the RMP system with the feedback control scheme of Theorem 1. Both the initial and the final desired values of these components are 0 rad/s.

Return to: Dynamics and Balance Control of the Reaction Mass Pendulum: A Three-Dimensional Multibody Pendulum With Variable Body Inertia

Copyright in the material you requested is held by the American Society of Mechanical Engineers (unless otherwise noted). This email ability is provided as a courtesy, and by using it you agree that you are requesting the material solely for personal, non-commercial use, and that it is subject to the American Society of Mechanical Engineers' Terms of Use. The information provided in order to email this topic will not be used to send unsolicited email, nor will it be furnished to third parties. Please refer to the American Society of Mechanical Engineers' Privacy Policy for further information.

Shibboleth is an access management service that provides single sign-on protected resources.
It replaces the multiple user names and passwords necessary to access subscription-based content with a single user name and password that can be entered once per session.
It operates independently of a user's location or IP address.
If your institution uses Shibboleth authentication, please contact your site administrator to receive your user name and password.